Page 73 - Applied statistics and probability for engineers

Section 2-6/Independence 51

Interestingly, P(B), the unconditional probability that the second part se lected is defective, without any knowl-
edge of the i rst part, is the same as the probability that the i rst part selected is defective. Yet our goal is to assess
independence. Because P(B | A) does not equal P(B), the two events are not independent, as we expected.

When considering three or more events, we can extend the deinition of independence with

the following general result.
Independence
(multiple events)
The events E , E , … ,E are independent if and only if for any subset of these events
1 2 n
P E i 1 ( ∩ E i 2 ∩ … ∩ E i k ) = ( ) × ( ) × … × P E i k ( ) (2-14)
P E i 1
P E i 2
This dei nition is typically used to calculate the probability that several events occur, assuming
that they are independent and the individual event probabilities are known. The knowledge that the
events are independent usually comes from a fundamental understanding of the random experiment.

Example 2-32 Series Circuit The following circuit operates only if there is a path of functional devices from
left to right. The probability that each device functions is shown on the graph. Assume that devices
fail independently. What is the probability that the circuit operates?
0.8 0.9

Let L and R denote the events that the left and right devices operate, respectively. There is a path only if both oper-
ate. The probability that the circuit operates is
(
P L ∩
P L and R) = ( R) = ( ) ( 0 80 . ) = .72
P L P R) = . (0
90
0
Practical Interpretation: Notice that the probability that the circuit operates degrades to approximately 0.5 when all
devices are required to be functional. The probability that each device is functional needs to be large for a circuit to
operate when many devices are connected in series.
Example 2-33 Semiconductor Wafers Assume that the probability that a wafer contains a large particle of con-
tamination is 0.01 and that the wafers are independent; that is, the probability that a wafer contains
a large particle does not depend on the characteristics of any of the other wafers. If 15 wafers are analyzed, what is the
probability that no large particles are found?
2
Let E i denote the event that the ith wafer contains no large particles, i = 1 , , … ,15 . Then, P E i ) = .0 99 The prob-
(
.
(
ability requested can be represented as P E 1 ∩ E 2 ∩ … ∩ E 15). From the independence assumption and Equation 2-14,
(
P E 1) × ( ) × …
P E 1 ∩ E 2 ∩ … ∩ E 15) = ( P E 2 × ( 0 99 = .
P E 15) = .
15
0 86
Example 2-34 Parallel Circuit The following circuit operates only if there is a path of functional devices from
left to right. The probability that each device functions is shown on the graph. Assume that devices
fail independently. What is the probability that the circuit operates?
0.95
a b

0.95
Let T and B denote the events that the top and bottom devices operate, respectively. There is a path if at least one
device operates. The probability that the circuit operates is